Sorry not on NCL. On NCL they only charge for evenings after 10:00 p.m. and on port days.
Carnival offers sitting for babies on port mornings until noon.
Disney would likely accomodate your needs the best of all the lines at this time.

Once baby turns two, Carnival would be great because baby will be able to attend the kids club anytime it is open and they will change diapers. NCL would be my next choice for a two year old but they will beep you to come change your kiddos nappies, which is not always conducive to a romantic dinner, but better than nothing.
We started cruising with our kids when the youngest was two, and to be honest it doesn't get much better than that. Its a wonderful thing for a family with small kids to do.

Yes, that's correct regarding Beenie Weenie's post -- NCL doesn't have private babysitting. They have group babysitting, which is an extension of the youth program after 10 p.m. for an hourly fee. However, your child must be two years old for that.